The effects of random fluctuations in an ocean current on underwater cw sound transmission between a bottomed source and receiver are determined for an ocean channel with a linear, depth‐dependent sound speed. A horizontal, depth‐dependent current is considered whose components are random processes. Effects of such a current on ray geometry are determined and six basic current‐induced ray states are found. Under certain conditions, including the assumption that the sound‐speed gradient is larger than current‐component gradients, only one ray state may arise. The geometry of this state is expressed explicitly in terms of the current. Approximations for travel times, total‐field intensity, and their first and second moments are obtained. These moments depend significantly on properties of the source–receiver current component. Intensity moments are predicted using ocean‐current data. For selected parameter values, a difference between relative mean intensity and relative intensity without current of as much as 9 dB and a standard deviation of relative intensity as large as 3 dB are found. These moments are rapidly varying with transmission range; however, useful bounds are derived which are slowly varying and which display an unusual behavior near certain critical ranges.